Solve 0=2e^frac{1{2}x}-1

$$0 = 2 e ^ { \frac { 1 } { 2 } x } - 1$$

Solve for x

$$x=-2\ln(2)\approx -1.386294361$$

Steps Using Definition of Logarithm

Swap sides so that all variable terms are on the left hand side.

$$2e^{\frac{1}{2}x}-1=0$$

Add 1 to both sides of the equation.

$$2e^{\frac{1}{2}x}=1$$

Divide both sides by 2.

$$e^{\frac{1}{2}x}=\frac{1}{2}$$

Take the logarithm of both sides of the equation.

$$\log(e^{\frac{1}{2}x})=\log(\frac{1}{2})$$

The logarithm of a number raised to a power is the power times the logarithm of the number.

$$\frac{1}{2}x\log(e)=\log(\frac{1}{2})$$

Divide both sides by \log(e).

$$\frac{1}{2}x=\frac{\log(\frac{1}{2})}{\log(e)}$$

By the change-of-base formula \frac{\log(a)}{\log(b)}=\log_{b}\left(a\right).

$$\frac{1}{2}x=\log_{e}\left(\frac{1}{2}\right)$$

Multiply both sides by 2.

$$x=-\frac{\ln(2)}{\frac{1}{2}}$$

Solve for x (complex solution)

$$x=-2\ln(2)+4\pi n_{1}i$$

$$n_{1}\in \mathrm{Z}$$